Circle Formulas — Complete Reference
What is Pi (π)?
Pi (π) is one of the most famous constants in mathematics. It is defined as the ratio of a circle’s circumference to its diameter: π = C/d. This ratio is the same for every circle, regardless of size. Pi is an irrational number, meaning its decimal representation neither terminates nor repeats: π = 3.14159265358979323846…
The history of π stretches back thousands of years. Ancient Babylonians used 3.125, and the Rhind Mathematical Papyrus (c. 1650 BC) gives a value of about 3.16. The Greek mathematician Archimedes of Syracuse (c. 287–212 BC) was the first to rigorously bound π, proving that 3 10/71 < π < 3 1/7 (i.e. 3.1408 < π < 3.1429) by inscribing and circumscribing polygons around a circle. By the 17th century, mathematicians were computing π to dozens of digits. Today, computers have calculated π to over 100 trillion decimal places.
For GCSE maths, use either π = 3.14 or the π button on your calculator for full accuracy. Exam answers are often given in terms of π (e.g., 9π cm²) to avoid rounding errors.
Parts of a Circle
Understanding the terminology is essential for tackling GCSE circle questions:
- Centre: The fixed point equidistant from all points on the circle.
- Radius (r): The distance from the centre to any point on the circumference.
- Diameter (d): A chord that passes through the centre; d = 2r.
- Circumference: The total length around the circle (its perimeter).
- Chord: A straight line segment connecting two points on the circumference. The diameter is the longest chord.
- Arc: A curved portion of the circumference. A minor arc is shorter than a semicircle; a major arc is longer.
- Sector: The region bounded by two radii and an arc (like a pizza slice). A minor sector contains the minor arc; a major sector contains the major arc.
- Segment: The region between a chord and the arc it cuts. Not to be confused with sector.
- Tangent: A straight line that touches the circle at exactly one point, perpendicular to the radius at that point.
- Secant: A line that intersects the circle at two points.
Deriving the Area Formula
The formula A = πr² can be derived by dividing a circle into thin concentric rings. Each ring of radius x and thickness dx has area approximately 2πx dx. Integrating from 0 to r: ∫₀⊃r; 2πx dx = πr². A more intuitive approach is to cut the circle into thin sectors (like pizza slices) and rearrange them into a rough rectangle. As the slices get thinner, the rectangle approaches dimensions πr (length = half circumference) × r (height = radius), giving area = πr².
Worked Examples
Example 1: Area and circumference from radius
A circular pizza has a radius of 15 cm. Find its area and circumference.
Area = π × 15² = 225π ≈ 706.86 cm²
Circumference = 2 × π × 15 = 30π ≈ 94.25 cm
Example 2: Sector area and arc length
A sector has radius 12 cm and central angle 150°. Find the arc length and sector area.
Arc length = (150/360) × 2 × π × 12 = (5/12) × 24π = 10π ≈ 31.42 cm
Sector area = (150/360) × π × 144 = (5/12) × 144π = 60π ≈ 188.50 cm²
Example 3: Find radius from circumference
A circular running track has a circumference of 400 m. What is the radius?
C = 2πr, so r = C / (2π) = 400 / (2 × 3.14159) = 400 / 6.2832 ≈ 63.66 m
Real-World Applications of Circles
Circles appear throughout everyday life and across many professional fields:
- Wheels and engineering: Every wheel, gear, pulley, and bearing is based on the circle. The circumference determines how far a vehicle travels per revolution; a tyre with circumference C travels C metres per rotation.
- Pipes and plumbing: The cross-sectional area of a pipe (A = πr²) determines its flow capacity. Doubling the radius quadruples the area and thus the potential flow rate.
- Architecture and design: Domes, arches, roundabouts, clock faces, and stadium layouts rely on circular geometry. The Pantheon in Rome features a perfectly circular opening (oculus) in its dome.
- Food and packaging: Pizza sizes, tin can dimensions, and cylindrical packaging all require circle calculations. A 14-inch pizza has approximately 154 in² of area versus a 12-inch’s 113 in² — a 36% difference that often surprises people.
- GPS and satellite coverage: A satellite’s coverage area on the Earth’s surface is approximately circular. The coverage area = πr² where r depends on the satellite’s altitude and beam angle.
- Astronomy: Planetary orbits are approximately circular (more accurately elliptical). Kepler’s laws and orbital mechanics heavily use π.
Circle Theorems for GCSE
Circle theorems are geometric rules about angles and lines in circles. They are tested at GCSE (Higher tier) and must be stated when used in exam proofs:
Circles in Coordinate Geometry
A circle with centre (a, b) and radius r has the equation:
For a circle centred at the origin: x² + y² = r². This is tested at both GCSE Higher and A-Level. You may be asked to:
- Find the equation of a circle given its centre and radius.
- Determine whether a point lies inside, on, or outside a circle.
- Find the intersection of a line and a circle by solving simultaneous equations.
- Find the equation of a tangent to a circle at a given point (using the fact that the tangent is perpendicular to the radius).
At A-Level, the general form x² + y² + 2gx + 2fy + c = 0 can be rewritten by completing the square to find centre (−g, −f) and radius √(g² + f² − c).
Frequently Asked Questions
What is the formula for the area of a circle?
The area of a circle = πr², where r is the radius. If you know the diameter d, use A = π(d/2)² = πd²/4. For example, radius = 5 cm gives area = 25π ≈ 78.54 cm².
How do you find the circumference of a circle?
Circumference = 2πr = πd. For radius 7 cm: C = 2 × π × 7 = 14π ≈ 43.98 cm. To find radius from circumference: r = C ÷ (2π).
What is π (pi)?
Pi (π) is the ratio of a circle’s circumference to its diameter, approximately 3.14159. It is irrational — its decimal never terminates or repeats. Archimedes first rigorously bounded π around 250 BC. Use the π button on your calculator for exam accuracy.
What is the difference between a chord and an arc?
A chord is a straight line connecting two points on the circumference. The diameter is the longest chord. An arc is the curved part of the circumference between those same two points. Arc length = (θ/360) × 2πr.
How do you calculate sector area?
Sector area = (θ/360) × πr², where θ is the central angle in degrees. For radius 6 cm and angle 90°: Area = (90/360) × π × 36 = 9π ≈ 28.27 cm². A semicircle (180°) has area = πr²/2.
What are circle theorems?
Circle theorems are GCSE geometry rules: (1) Angle at centre = twice angle at circumference. (2) Angles in same segment are equal. (3) Angle in semicircle = 90°. (4) Opposite angles in a cyclic quadrilateral sum to 180°. (5) Tangent is perpendicular to radius. These must be stated by name in exam proofs.